Write the n x n zero matrix and the n x n identity matrix (denoted I_n).

Create the n x n zero and identity matrices in MATLAB using the ZERO and EYE commands.

Determine whether two matrices are equal.

Add two matrices of equal size.

Calculate a scalar multiple of a matrix.

Use Theorem 2.1 to calculate sums and scalar multiples of matrices.

Determine whether two matrices can be multiplied together. If they can, carry out this multiplication by hand and using MATLAB.

Carry out matrix multiplication using the definition and using the row-column rule.

State the properties of matrix multiplication as given in Theorem 2.2.

Raise a matrix to a power.

Form the transpose of a matrix. (Also be able to do this in MATLAB.)

EXERCISES FROM LAY: 1--9, 17--20, 39, 40. ALSO -- Mastery Exam MO will assess your skill at these competency areas.

2.2: The Inverse of a Matrix

State the definition of the inverse of a matrix and explain why we cannot think of it as "division".

Use the term "nonsingular" to refer to an invertible matrix and "singular" to refer to a non-invertible matrix.

Determine whether a 2x2 matrix is invertible by calculating its determinant. If the matrix is nonsingular, calculate its inverse using Theorem 2.4.

Solve a system of linear equations using a matrix inverse as in Theorem 2.5.

State and use the properties of matrix inverses as in Theorem 2.6.

Give full proofs of the matrix properties in Theorem 2.6, perhaps for a special case (for example, n = 2 or n = 3).

State the definition of an elementary matrix, and write out an elementary matrix of a given size corresponding to a specific row operation.

Explain how repeated multiplication of elementary matrices corresponds to row-reduction.

Find the inverse of an elementary matrix.

State Theorem 2.7 and use its result to find the inverse of an nxn matrix (or to determine that such a matrix is singular).

EXERCISES FROM LAY: 1--7, 9, 17, 18, 21--24, 29--33.

2.3: Characterizations of Invertible Matrices

State all 12 parts of the Invertible Matrix Theorem as given in this section.

Explain the logical equivalency of any two parts of the Invertible Matrix Theorem. (Being able to prove that logical equivalency is an excellent way to explain it.)

EXERCISES FROM LAY: 1--10 (don't just calculate A^(-1) in MATLAB; use the results of the IVT, for example by determining that the columns of the matrix are linearly dependent or span all of R^3); 11, 13, 15--24.

2.5: Matrix Factorizations

Define what an LU factorization (sometimes called an LU "decomposition") of a matrix is.

Explain why an LU factorization for a matrix is useful.

Use the tic and toc commands in MATLAB to find the time elapsed during a MATLAB calculation and use these commands to compare the time needed to solve a linear system using "straight" row reduction versus using an LU factorization.

Find an LU factorization of a matrix by hand.

Find an LU factorization of a matrix using the LU command in MATLAB.

Solve a linear system using an LU factorization.

EXERCISES FROM LAY: 1, 3, 5, 7--18.

2.8: Subspaces of Rn

State the definition of a subspace of Rn.

Determine whether a given subset of Rn is or isn't a subspace of Rn.

State the definitions of the column space and null space of a matrix.

Determine whether a given vector is in the column space of a matrix.

Determine whether a given vector is in the null space of a matrix.

State the definition of a basis for a subspace of Rn.

Find a basis for the column space of a matrix.

Find a basis for the null space of a matrix.

EXERCISES FROM LAY: 3, 5 ("generated" means "spanned"), 7, 9, 11, 13, 15, 17, 19, 21, 23, 25.

2.9: Dimension and Rank

Find the coordinate vector for a given vector relative to a given basis.

State the definition of the dimension of a subspace of Rn.

State the definition of the rank of a matrix.

Find the dimension of a subspace of Rn.

Find the rank of a given matrix.

State the Rank Theorem (Theorem 2.14).

State the Basis Theorem (Theorem 2.15).

State the six new parts of the Invertible Matrix Theorem.

Calculate the determinant of a 2x2 matrix by hand.

Calculate the determinant of a 3x3, 4x4, etc. matrix by hand, using an expansion along any row or any column. Make intelligent choices of which column/row to use, and get the sign pattern right.

Find the determinant of an upper- or lower-triangular matrix without resorting to the full definition of the determinant. (See Theorem 3.2)

EXERCISES FROM LAY: 1, 5, 9, 13, 25, 27, 39.

3.2: Properties of Determinants

Given the determinant of a matrix A, find the determinant of the matrix obtained from A by any of the three elementary row operations. (See Theorem 3.3)

Given the determinant of a matrix A, state whether or not A is invertible.

Given the determinant of a matrix A, draw conclusions about the properties of A based on the Invertible Matrix Theorem. (For example, state whether the columns of A are linearly independent, what the rank of A is, etc.)

Given the determinant of a matrix A, state the determinant of A^T (the transpose of A).

Given the determinant of matrices A and B, state the determinant of AB.

Given the determinant of an invertible matrix A, state the determinant of A^{-1}.

EXERCISES FROM LAY: 5, 9, 15--25 odd, 29 (do this without actually calculating B^5), 35, 39.

Coverage for In-Class Assessment 2 includes all of the above from Chapters 2 and 3.

Chapter 5: Eigenvalues and Eigenvectors

5.1: Eigenvectors and Eigenvalues

State the definitions of eigenvector, eigenvalue, and eigenspace.

Determine if a given vector is or is not an eigenvector for a matrix.

Given an eigenvalue for a matrix, find an eigenvector (and a basis for the eigenspace) corresponding to it.

Find (quickly) the eigenvectors for a triangular matrix.

State and use the result of Theorem 5.2 about linear independence of eigenvectors corresponding to different eigenvalues.

EXERCISES FROM LAY: 1--17 odd, 21, 23, 31, 37, 39. (The appropriate MATLAB command for 37 and 39 is eig, but you need to read the documentation on that command to see what it actually does.)

5.2: The Characteristic Equation

Find the characteristic polynomial for a square matrix.

Use the characteristic polynomial for a matrix to set up the characteristic equation, then solve the characteristic equation to find eigenvalues for a matrix.

EXERCISES FROM LAY: 1--17 odd (read in the text to learn about "multiplicities"), 21.

5.3: Diagonalization

Raise a diagonal matrix to a power without doing any matrix multiplication.

State what it means for two matrices to be "similar".

Raise a matrix to a power without doing more than two matrix multiplications, if it is known that the matrix is similar to a diagonal matrix.

Determine whether or not a matrix is diagonalizable.

Diagonalize a matrix if it is determined that the matrix is diagonalizable.

EXERCISES FROM LAY: 1--21 odd, 33, 35.

Chapter 6: Orthogonality and Least Squares

6.1: Inner Product, Length, and Orthogonality

Calculate the inner (dot) product of two vectors.

State and use the properties of the inner product as given in Theorem 6.1.

Prove one or more of the properties of the inner product as given in Theorem 6.1.

Calculate the length ("norm") of a vector.

Determine whether a vector is a unit vector.

Given a nonzero vector, find a unit vector pointing in the same direction. (I.e., "normalize" the vector.)

Find the distance between two vectors.

Determine whether two vectors are orthogonal.

State the Pythagorean Theorem in terms of vector norms (Theorem 6.2).

EXERCISES FROM LAY: 1--17 odd, 25, 27.

6.2: Orthogonal Sets

Determine whether a set of vectors forms an orthogonal set.

Give a partial proof of Theorem 6.4 (orthogonal sets are linearly independent).

Determine whether a set of vectors is an orthogonal basis for a given subspace.

Given an orthogonal basis and a vector in the space spanned by that basis, use Theorem 6.5 to find the weights in a linear combination of basis vectors to get the given vector.

Calculate the projection of a vector onto a line.

Determine whether a set of vectors is orthonormal (or, whether a set of vectors forms an orthonormal basis for a subspace).

Determine whether a matrix has orthonormal columns using Theorem 6.6.

State and use the properties of matrices with orthonormal columns as given in Theorem 6.7.

EXERCISES FROM LAY: 1--21 odd, 27.

6.3: Orthogonal Projections

Define the concept of the orthogonal complement to a subspace. (This is actually found in section 6.2 of the textbook.)

Given a subspace W of Rn and a vector y in Rn (NOT necessarily in W), write y as a sum of two vectors -- one of which is in W and the other of which is orthogonal to W. (This uses the formula in the Orthogonal Decomposition Theorem.)

If a vector y belongs to a subspace W, find (quickly!) the projection of y into W.

Calculate the distance from a vector y to a subspace W.

State Theorem 6.9, the Best Approximation Theorem, and explain what it means in plain English.

Given an orthonormal basis for a subspace W of Rn and vector y in Rn, find proj_L (y) using the simplified formula in Theorem 6.10.

EXERCISES FROM LAY: 1--17 odd, 21.

6.4: The Gram-Schmidt Process

Be able to replicate and/or explain Example 2 in the textbook.

Explain the purpose of the Gram-Schmidt process and give a general outline of how it works.

Use the Gram-Schmidt process to take a given basis for a subspace of Rn and construct an orthogonal basis for the same subspace.

## Section-by-section competencies for MAT 233, Linear Algebra

## Chapter 1

The review page for In-Class Assessment 1 serves as the competency list for this chapter.## Chapter 2: Matrix Algebra

2.1: Matrix OperationsZEROandEYEcommands.EXERCISES FROM LAY: 1--9, 17--20, 39, 40. ALSO -- Mastery Exam MO will assess your skill at these competency areas.2.2: The Inverse of a MatrixEXERCISES FROM LAY: 1--7, 9, 17, 18, 21--24, 29--33.2.3: Characterizations of Invertible MatricesEXERCISES FROM LAY: 1--10 (don't just calculate A^(-1) in MATLAB; use the results of the IVT, for example by determining that the columns of the matrix are linearly dependent or span all of R^3); 11, 13, 15--24.2.5: Matrix Factorizationsticandtoccommands in MATLAB to find the time elapsed during a MATLAB calculation and use these commands to compare the time needed to solve a linear system using "straight" row reduction versus using an LU factorization.LUcommand in MATLAB.EXERCISES FROM LAY: 1, 3, 5, 7--18.2.8: Subspaces of RnEXERCISES FROM LAY:3, 5 ("generated" means "spanned"), 7, 9, 11, 13, 15, 17, 19, 21, 23, 25.2.9: Dimension and RankEXERCISES FROM LAY:1--13 odd, 17, 19, 21, 23, 25, 29.## Chapter 3: Determinants

3.1: Introduction to DeterminantsEXERCISES FROM LAY: 1, 5, 9, 13, 25, 27, 39.3.2: Properties of DeterminantsEXERCISES FROM LAY: 5, 9, 15--25 odd, 29 (do this without actually calculating B^5), 35, 39.Coverage for In-Class Assessment 2 includes all of the above from Chapters 2 and 3.Chapter 5: Eigenvalues and Eigenvectors

5.1: Eigenvectors and Eigenvalues: 1--17 odd, 21, 23, 31, 37, 39. (The appropriate MATLAB command for 37 and 39 isEXERCISES FROM LAYeig, but you need to read the documentation on that command to see what it actually does.)5.2: The Characteristic Equation5.3: Diagonalization## Chapter 6: Orthogonality and Least Squares

6.1: Inner Product, Length, and Orthogonality6.2: Orthogonal Sets6.3: Orthogonal Projections6.4: The Gram-Schmidt Process6.5: Least-Squares Problems